Jesus declared that there would be a time where the disciples could not be with Jesus in person. Thus, Jesus took the time with the disciples to share some important things.
One of them was a new commandment (as in new in development) -- "love one another" similar to how Jesus loved them (13:34). Jesus sets the example of love -- washing feet in service, guiding the disciples, and later sacrificing himself to take on the penalty of sin on behalf of humanity. Love is one of the biggest fundamentals of the Bible, and it's worth remembering that God loves us in spite of our screw-ups.
In fact, loving one another is how the surrounding people can know that the disciples are Jesus's disciples (13:35). God uses instances of love from the disciples to reveal Jesus to others. Even today, showing love to other people (even if they are different or have disagreements) could impact them -- and God still uses instances of Christ-like love today to reveal Jesus's character.